The graph structure of two-player games

TL;DR

This paper analyzes two-player games through response graphs, revealing structural properties and characterizations related to zero-sum and potential games, with implications for game dynamics and strategy dominance.

Contribution

It introduces a response graph model for two-player games, characterizes games sharing graphs with zero-sum or potential games, and explores their structural and strategic implications.

Findings

Response graphs capture key properties of strategic interactions.

Games sharing response graphs with zero-sum and potential games are dominance-solvable.

Every non-iteratively-dominated strategy appears in subgames with specific graph structures.

Abstract

In this paper we analyse two-player games by their response graphs. The response graph has nodes which are strategy profiles, with an arc between profiles if they differ in the strategy of a single player, with the direction of the arc indicating the preferred option for that player. Response graphs, and particularly their sink strongly connected components, play an important role in modern techniques in evolutionary game theory and multi-agent learning. We show that the response graph is a simple and well-motivated model of strategic interaction which captures many non-trivial properties of a game, despite not depending on cardinal payoffs. We characterise the games which share a response graph with a zero-sum or potential game respectively, and demonstrate a duality between these sets. This allows us to understand the influence of these properties on the response graph. The response graphs of Matching Pennies and Coordination are shown to play a key role in all two-player games: every non-iteratively-dominated strategy takes part in a subgame with these graph structures. As a corollary, any game sharing a response graph with both a zero-sum game and potential game must be dominance-solvable. Finally, we demonstrate our results on some larger games.